Minimum disparity estimation in controlled branching processes

Minimum disparity estimation in controlled branching processes is dealt with by assuming that the offspring law belongs to a general parametric family. Under some regularity conditions it is proved that the minimum disparity estimators proposed -based on the nonparametric maximum likelihood estimator of the offspring law when the entire family tree is observed- are consistent and asymptotic normally distributed. Moreover, it is discussed the robustness of the estimators proposed. Through a simulated example, focussing on the minimum Hellinger and negative exponential disparity estimators, it is shown that both are robust against outliers, being the negative exponential one also robust against inliers.


Introduction
Branching processes are useful models for the description of the dynamics of systems whose elements produce new ones following probability laws. Its theory has been developed from simple models to increasing realism. Added to the theoretical interest in these processes there is therefore a major practical dimension due to their potential applications in such diverse fields as biology, epidemiology, genetics, medicine, nuclear physics, demography, actuarial mathematics, algorithm and data structures, see, for example, the monographs Devroye (1998), Haccou, Jagers, and Vatutin (2005), González, Martínez, Molina, Mota, Puerto, and Ramos (2010) and Kimmel and Axelrod (2015).
In particular, controlled branching processes (CBPs) are discrete time stochastic processes very appropriate to describe the growth of populations in which the number of participating individuals in the reproduction process is determined in each generation by a control mechanism. Besides, as is common in the branching framework, every individual reproduces independently of the others following *Corresponding author. Phone 0034924289300. Ext. 86820 the same probability law, which is called the offspring distribution.
The versatility of these models makes possible to model different kind of migratory movements. Thus, several well-known branching processes can be included in this class as particular cases, for instance, the own Bienaymé-Galton-Watson process, the branching processes with immigration (see Sriram (1994) and references therein), with random migration (see Yanev and Yanev (1996)), with immigration at state zero (see Bruss and Slavtchova-Bojkova (1999)) or with bounded emigration (see del Puerto and Yanev (2008)). Other interesting particular cases are branching processes with adaptive control (see Bercu (1999)) or with continuous state space (see Rahimov and Al-Sabah (2007)).
Since the appearance of the pioneering publication by Yanev (1975), its probability theory has been widely studied. The development of its inference theory, which guarantees the applicability of these models, has become the main goal in the most recent researches. For this issue, in a frequentist framework, it is important to mention Dion and Essebbar (1995), Mohan (2000), del Puerto, 2004, 2005a), Sriram, Bhattacharya, González, Martínez, and del Puerto (2007), and González, Minuesa, and del Puerto (2016). From a Bayesian standpoint, one can find the papers Martínez, Mota, and del Puerto (2009) and González, Gutiérrez, Martínez, and del Puerto (2013). It is well-known that, in general, the estimators based on maximizing the likelihood function are badly affected by outliers. This also happens in CBP context, as is pointed out in the simulated example at the end of the paper, reason why robust procedures must be developed.
Robust estimation has barely developed in the context of branching processes. One only can find results in the frame of the Bienaymé-Galton-Watson processes, by using weighted least trimmed estimation (see Stoimenova, Atanasov, and Yanev (2004)) and by considering minimum Hellinger distance estimation (see Sriram and Vidyashankar (2000)). Our main aim in this paper is to carry out, in a frequentist framework, robust estimation for the general class of CBPs.
To this end, we make use of the minimum disparity methodology. This methodology has arisen as one that attains robustness properties without loss of efficiency.
It was introduced in Lindsay (1994) for discrete models and since then, the literature on it has experimented a large growth (see Pardo (2006) and Basu, Shioya, and Park (2011) for further information). In our context, assuming that the offspring distribution belongs to a very general parametric family, we determine minimum disparity estimators (MDEs) of the underlying parameter and study their asymptotic and robustness properties. The method consists of proposing as estimator of the offspring distribution that parametric distribution which minimizes the discrepancy with a nonparametric estimator based on some observed sample. The discrepancy is measured by a function called disparity measure.
Thus, one can obtain different MDEs depending on the nonparametric estimator and the disparity measure considered. Special interest is highlighted in this paper for the negative exponential disparity and the Hellinger distance. The maximum likelihood estimator based on the observation of the whole family tree until a certain generation is considered as the nonparametric estimator. This paper presents for the first time the application of the technique of minimum disparity for the general class of branching structure given by CBPs, hence extending the results in Sriram and Vidyashankar (2000) in a double sense: model and measure, and moreover extending the results in Basu, Sarkar, and Vidyashankar (1997) and Park and Basu (2004) from an independent and identically distributed (i.i.d.) and continuous context to a dependent and discrete setup. It is worthwhile to point out the fundamental roles played by the nonparametric estimator and the dependence structure of the CBP to obtain the asymptotic properties of the MDEs proposed, that require a different approach from those already established in the i.i.d. setting.
Besides the introduction, this paper is organized into 6 sections and an appendix. In Section 2 we present the formal model and establish some hypotheses that we assume throughout the paper. Section 3 is devoted to defining and describing minimum disparity estimation. The asymptotic properties of MDEs are also studied; to this end, we introduce the disparity functional associated to a disparity measure and research its properties. The robustness of MDEs is studied in Section 4. To illustrate this methodology, we present a simulated example in Section 5. Concluding remarks about the contributions of the paper are presented in Section 6. Finally, we dedicate an appendix to the proofs of the theorems, in order to facilitate the reading of the paper.

The probability model
We consider a controlled branching process with random control function (CBP). Mathematically, this process is a discrete-time stochastic model {Z n } n∈N defined recursively as: being N a nonnegative integer, {X nj : n = 0, 1, . . . ; j = 1, 2, . . .} and {φ n (k) : n, k = 0, 1, . . .} two independent families of nonnegative integer valued random variables. Moreover, X nj , n = 0, 1, . . ., j = 1, 2, . . ., are i.i.d. random variables and for each n = 0, 1, . . ., {φ n (k)} k≥0 , are independent stochastic processes with equal one-dimensional probability distributions. The empty sum in (2.1) is considered to be 0. We denote by p = {p k } k≥0 the common probability distribution of the random variables X nj , i.e., p k = P [X nj = k], k ≥ 0, which is known as offspring distribution or reproduction law, and by m and σ 2 its mean and variance (assumed finite), and we referred to them as offspring mean and variance, respectively. We also denote ε(k) = E[φ 0 (k)] and σ 2 (k) = V ar[φ 0 (k)] the mean and the variance of the control variables (assumed finite too).
In addition, we suppose that the offspring distribution belongs to a general parametric family: where p(θ) = {p k (θ)} k≥0 and Θ is a subset of R, that is, p = p(θ 0 ) for some θ 0 ∈ Θ, referred as to the offspring parameter. It is the aim of this paper to estimate θ 0 efficiently and robustly by choosing θ ∈ Θ which provides the best adjustment to the observed sample in terms of the disparity measures.
To develop this methodology we need to consider nonparametric estimators of the offspring distribution. In this sense, in González, Minuesa, and del Puerto (2016), nonparametric maximum likelihood estimators (MLEs) based on different samples are provided. Let denote a generic nonparametric estimator of p based on a sample, say X n , byp n = {p n,k } k≥0 , verifyingp n,k ≥ 0, for each k ≥ 0, and ∞ k=0p n,k = 1 (where n indicates that we observe the data up to the generation n).

Minimum disparity estimation
In this section, we introduce the notions of disparity measure and minimum disparity estimator, and present several interesting examples of them. Although we focuss our attention on probability distributions defined on the nonnegative integers, that is, those which can be offspring distributions, the definitions and results given in this section keep valid for whatever discrete model. Let Γ be the set of all probability distributions defined on the nonnegative integers, F Θ the parametric family introduced in (2.2), and G(·) a three times differentiable and strictly convex function on [−1, ∞) with G(0) = 0. The disparity measure ρ corresponding to G(·) is defined for any q ∈ Γ and θ ∈ Θ, as where δ(q, θ, k) denotes the "Pearson residual at k", that is, Notice that the Pearson residual at k depends on the probability distribution q and on the parameter θ, and that δ(q, θ, k) ∈ [−1, ∞), for each q ∈ Γ, θ ∈ Θ, and k ≥ 0.
Due to the fact that G(·) is strictly convex, one has that ρ is nonnegative.
Moreover, when G(·) is also nonnegative and has a unique zero at 0 it is verified that ρ(q, θ) = 0 if and only if q = p(θ). Given a sample X n and a nonparametric estimator of p,p n , based on it, we define the minimum disparity estimator (MDE) of θ 0 for the disparity measure ρ based onp n as θ ρ n (p n ) = arg min θ∈Θ ρ(p n , θ). (3.1) Remark 1. Some interesting cases of nonnegative disparity measures are the following: (a) The disparity obtained with the function G(δ) = (δ + 1) log(δ + 1) is a kind of the Kullback-Leibler divergence. It is denoted by LD(p n , θ), for eachp n , n ∈ N, and θ ∈ Θ, and it is known as likelihood disparity. Its minimizer,θ LD n (p n ), is known as the minimum likelihood disparity estimator (MLDE). In some cases, this estimator coincides with the MLE.
Other examples are the family of power divergence measures (see Cressie and Read (1984)), the blended chi-squared measures, which include Pearson's chi-squared and Neyman's chi-square, the blended weight chi-squared measures and the blended weight Hellinger distance family (see Lindsay (1994)).
Under conditions of differentiability of the model, a useful way for determining a MDE is to take into account that it must satisfyρ(p n ,θ ρ n (p n )) = 0, withρ(q, θ) denoting the first derivative of ρ(q, θ) with respect to θ, q ∈ Γ. It is verified, for q ∈ Γ and θ ∈ Θ, with A(δ) = (δ + 1)G ′ (δ) − G(δ), and G ′ (·) and p ′ k (·) denoting the first derivative of G(·) and p k (·), respectively. The function A(·) is called the residual adjustment function (RAF) of the disparity and it is an increasing function on [−1, ∞), and twice differentiable that can be redefined so that A(0) = 0 and A ′ (0) = 1.
(b) The RAF of the squared Hellinger distance after the standardization is (c) The RAF of the negative exponential disparity corresponding to In Lindsay (1994), the RAFs of different disparity measures are compared (see Figures 4 and 5 in Lindsay (1994)). The RAF of a disparity measure is relevant in determining the efficiency and robustness properties of the corresponding MDE. Concretely, A ′′ (0) is demonstrated to play a key role: large negative values of A ′′ (0) correspond to robustness properties and zero value matches a secondorder efficient estimator in the sense of Rao (1961).
Before focussing on these matters, first we establish the existence of the minimum in (3.1) and its uniqueness. To this end, we consider the disparity functional associated with a disparity ρ, defined as T ρ : Γ → Θ, with T ρ (q) = arg min θ∈Θ ρ(q, θ), whenever the minimum exists. Notice that there might exist multiple values of the parameter θ which minimize the function ρ(q, ·). As a consequence, T ρ (q) would denote any of these values. Moreover,θ ρ n (p n ) = T ρ (p n ). It is easy to show that if the paramater space Θ is compact, then the disparity functional is well defined for each q ∈ Γ such that the function ρ(q, ·) is continuous in Θ. However, we will weaken the compactness of Θ in a similar way as was done in Simpson (1987) and in Sriram and Vidyashankar (2000). Specifically, given a disparity ρ, we limit our study to the subclassΓ ρ ⊆ Γ which verifies the following condition: there exists a compact set C ρ ⊆ Θ such that for every q ∈Γ ρ , Theorem 1. It is satisfied: (i) For each q ∈Γ ρ verifying that ρ(q, ·) is continuous in C ρ , there exists T ρ (q).
Remark 3. (i) Notice that the continuity of p k (·) for each k ≥ 0 in an arbitrary set B ⊆ Θ leads to the continuity in B of the function ρ(q, ·) associated with any disparity measure ρ determined by a bounded function G(·), for each q ∈ Γ. This is deduced by a generalized dominated convergence theorem (see Royden (1988), p.92). The aforesaid condition is satisfied by the negative exponential disparity.
Although the Hellinger distance is defined by a nonbounded function G(·), in this case the condition of continuity of p k (·) in B for each k ≥ 0 is enough to obtain the continuity of HD(q, ·) in B for each q ∈ Γ. This latter is followed by the Cauchy-Schwarz inequality and the Scheffé's theorem.
(ii) The redefinition of some disparities, without affecting their minimizer, so that the related functions G(·) will be nonnegative and have a unique zero at 0 is possible. For instance, for the negative exponential disparity we can consider the functionḠ(δ) = G(δ) + δ instead of G(δ), which verifies the previous properties In order to study the asymptotic properties of the MDEs, we are to assume several conditions. Let fix the next assumptions: (A1) ρ is a disparity measure associated with a function G(·) which verifies that Notice that under (A2), by Theorem 1, T ρ (q) exists for every q ∈Γ ρ . LetΓ ρ be the set of q ∈Γ ρ such that T ρ (q) is unique. Now, in the following theorem the continuity of the disparity functional is established. Henceforth, all the limits are taken as n → ∞.
Theorem 2. Let q and {q n } n∈N be in Γ such that q n → q in l 1 . Assuming (A1), (A2) and that q ∈Γ ρ , then T ρ (q n ) eventually exists and the functional T ρ (·) is In the case of the Hellinger distance, despite condition (A1) is not satisfied by this disparity, one can establish the following result.
Recall that p = p(θ 0 ) is the true reproduction law. Observe that under (A1), if (A2) is verified and p ∈Γ ρ , one obtains T ρ (p) = θ 0 and for the case of Hellinger distance, dropping (A1), one also has T HD (p) = θ 0 . Next theorem establishes the strong consistency of the MDEs.
Theorem 4. Assume (A2) and p ∈Γ ρ , for the corresponding disparity ρ. Then under conditions which guarantee thatp n,k is a strongly consistent estimator of p k (θ 0 ), for each k ≥ 0, one has that: The proof can be consulted in Appendix.

Asymptotic efficiency
The results given in the previous section are general in the sense that the explicit expression of the nonparametric estimator is not required, and one only needs to know its properties, as for example, its consistency. However, to establish the asymptotic efficiency of the MDEs, explicit formulas of the nonparametric estimators are needed. Then, to develop this section we come back to the CBP context. In González, Minuesa, and del Puerto (2016), we give nonparametric estimators of the offspring distribution under several sampling schemes. In particular, in a complete data context, we consider the entire family tree up to generation n can be observed, that is, the sample Z González, Minuesa, and del Puerto (2016)), is given by p n = { p n,k } k≥0 : It is proved that p n,k is strongly consistent for p k on {Z n → ∞}, for each k ≥ 0, (see Theorem 3.6 in González, Minuesa, and del Puerto (2016)), under the following assumption: (A3) Let consider a CBP satisfying that: Remark 4. For CBPs verifying (a), sufficient conditions for (b), (c) and (d) hold are discussed in González, Minuesa, and del Puerto (2016).
As a consequence, under (A3), from Theorem 4 (i) and (ii), one obtains, respectively, that the estimatorsθ ρ n (p n ) andθ HD n (p n ) are strongly consistent on {Z n → ∞}. Now, we focus our attention on the asymptotic efficiency. To this end, we must consider additional conditions on the functions p(·). We assume from now on that for each k ≥ 0, p k (θ) is twice continuously differentiable with respect to θ and: Remark 5. Notice that for a disparity ρ satisfying (A5), (A4) is a sufficient condition to guarantee that ρ(q, θ) can be twice differentiable with respect to θ.
It is easy to check that the negative exponential disparity satisfies (A5) but the Hellinger distance does not. Thus, to establish the efficiency of MHDE instead of the previous hypotheses we will assume the following condition on ) in a similar way to that in Beran (1977): in l 2 , that is, there exist s ′ (θ) ∈ l 2 and s ′′ (θ) ∈ l 2 verifying that for every β in a neighbourhood of zero In addition, observe that although conditions (A1) and (A5) seem to be quite restrictive, they are satisfied by a wide set of disparities (see Park and Basu (2004)).

Robustness
In this section, we address the issue of the robustness of the MDEs of θ.
For this purpose, we will study the behaviour of the corresponding disparity functional under contamination by considering the following model: where α ∈ (0, 1), θ ∈ Θ, L ∈ N 0 and η L is a point mass distribution at a nonnegative integer L. This model is called mixture model for gross errors at L and it represents the simplest context of contamination. This approach was introduced in Tukey (1960) and consists of assuming the contaminated model instead of the model distribution in order to explain or incorporate the outliers.
In the analysis of robustness of an estimator, an essential tool is the influence curve, which for each disparity ρ is a function of L ∈ N 0 defined as lim α→0 α −1 (T ρ (p(α, θ, L)) − T ρ (p(θ))).
Although the unboundedness of this function is an indicator of the misbehaviour of the MDEs of θ in presence of outliers, the influence curve can be very a deceptive measure of robustness (see Lindsay (1994)). For this reason, we will also examine the α-influence curves of T ρ (·), which are functions of L ∈ N 0 defined as α −1 (T ρ (p(α, θ, L)) − T ρ (p(θ))), for each α ∈ (0, 1).
Next theorem, whose proof can be read in Appendix, provides an expression for the influence curves and establishes conditions under which the disparity functional is robust at p(θ) against 100α% contamination by gross error at an arbitrary integer.
(b) T ρ (p(θ, α, L)) is a bounded and continuous function of L. (i-c) are also satisfied.
Another important concept in the study of the robustness is the breakdown point. The breakdown point of a disparity functional T ρ (·) at q ∈ Γ is given by: Intuitively, the breakdown point represents the smallest amount of contamination that can cause the estimator to take arbitrarily large values. Note that b(α; T ρ , q) = ∞ is equivalent to the existence of a sequence of probability distributions {q n } n∈N verifying |T ρ ((1 − α)q + αq n ) − T ρ (q)| → ∞ and in that case, we say there is breakdown in T ρ (·) for a level of contamination equals α. The sequence {q n } n∈N is called sequence of contaminating probability distributions. This fact is useful for the establishment of a lower bound for the breakdown point of the MDEs of θ in the following theorem.
Theorem 7. (i) Assume that the contaminant sequence {q n } n∈N , the distributions of the family F Θ , and θ * ∈ Θ verify: Then, the asymptotic breakdown point of the MDE of θ * is at least 1/2.
The proof of (i) is analogous to those given in Theorem 4.1 in Park and Basu (2004) replacing integrals with sums. The proof of (ii) is exactly the same as Theorem 3 in Simpson (1987) and it is omitted. In particular, since p = p(θ 0 ), then̺ = 1 and as a consequence, the asymptotic breakdown point for HD is at least 1/2 when ̺ * = 0, which usually holds.

Simulated example
Through a simulated example, we compare the behaviour of the MHDEs, MNEDEs and MLDEs based on the whole family tree under an uncontaminated model and under mixture models for gross errors. To this end, we have considered as initial model a CBP starting with one individual and Poisson distributions as offspring and control distributions. In particular, the offspring distribution is a Poisson distribution with the parameter θ 0 = 7 and the variable φ n (k) follows Poisson distribution with parameter λk, with λ = 0.3, for each k ≥ 0, n ≥ 0.
Therefore, the offspring mean and variance are m = σ 2 = 7, and τ m = θ 0 λ = 2.1 (see (A3) for definition). The parameter τ m is referred to as the asymptotic mean growth rate and is the threshold parameter of the model (see González, Molina, and del Puerto (2005b)). In practice, control functions, φ n (k), following Poisson distributions of parameters λk are appropriate to describe an environment with expected immigration or emigration depending on λ > 1 or < 1. In our example we consider a model with expected emigration although supercritical (τ m > 1).
First, we show that in a contamination-free context, MHDEs and MNEDEs are as efficient as MLDEs. To this end, we have simulated 10 generations of N = 100 CBPs following the previous model, and we have estimated the relative efficiency ofθ N ED n (p n ) toθ HD n (p n ), ofθ HD n (p n ) toθ LD n (p n ) and ofθ N ED n (p n ) tõ θ LD n (p n ) in each generation by the ratios of these mean squared errors: For a generic CBP, the information given by a sample observed until a fixed generation n depends on its asymptotic mean growth rate, τ m ; being poorer when τ m ≈ 1 than when τ m > 1. This implies that to compare the behaviour of the different estimators for each one of the contaminated models (which have asymptotic mean growth rates, called τ m (θ 0 , α, L), of different magnitudes) one needs to observe different numbers of generations depending on τ m (θ 0 , α, L)-going in Table 1: Relative bias for Hellinger distance and negative exponential disparity for the mixture models for gross errors with L = 0 and different values of α. In addition, Figure 1 (thrid row -right) shows the contour plot of the asymptotic mean growth rate of the contaminated models, τ m (θ 0 , α, L), with the underlying points representing the minimum disparity method which provides the smallest MSE for each contaminated model. In view of these plots, one can deduce that the MNEDE supplies more accurate estimates in most of the contaminated models (166 models, that is 63.85% of the models), but the best method when the contaminated state is between 3 and 11 is usually the MHD (85 models, that is 32.69% of models). However, the MLDE only behaves properly in 9 models (3.46% of the models), where L is equal 7 (consequently, the offspring mean remains unchanging). We have also studied the performance of these methods in presence of inliers, which correspond to the model introduced at (5.1) with α < 0 such that p(θ, α, L) is a probability distribution. To this end, we compare the potential bias, defined as ∆T ρ (α, L) = T ρ (p(θ 0 , α, L)) − T ρ (p(θ 0 )),  with ρ ∈ {N ED, HD, LD}. In fact, we examine the relative bias of MHDE and MNEDE with respect to MLDE under mixture model for gross errors located at L = 0 (Table 1), at L = 8 (Table 2) and at L = 20 (Table 3)

Concluding remark
In the context of controlled branching processes with random control functions, assuming a general parametric framework for the offspring distribution, we have studied the minimum disparity estimation of its main parameter.
First, we have established conditions for the existence and uniqueness of MDEs for a general discrete model. Moreover, it has been established that the proposed MDEs are strongly consistent as the associated nonparametric estimators are. In particular, we have considered as the nonparametric estimator of the offspring law, the MLE based on the observation of the entire family tree until a certain generation, which is consistent under some regularity conditions. Based on this nonparametric estimator, the limiting normality of the corresponding MDEs of the offspring parameter, suitably normalized, has been also established. These results are regarded as a generalization of those given for Bienaymé-Galton-Watson processes (see Sriram and Vidyashankar (2000)),  nonnegative and has a unique zero at 0, ρ(p(θ * ), θ) = 0 if and only if p(θ * ) = p(θ), and from the identifiability of F Θ , this can only occurs when θ * = θ.

Proof of Theorem 2
We present an adaptation and extension of the proofs of Proposition 2 in Basu, Sarkar, and Vidyashankar (1997) and of Theorem 3.2 in Park and Basu (2004), developed for general continuous models.
Thus, using Theorem 1, there exists T ρ (q n ), which we denote θ n to ease the notation, and θ n ∈ C ρ eventually. Finally, one has to show that θ n → θ.
(i) It is immediate from Theorem 2 and the fact thatp n → p a.s. in l 1 .
(ii) The proof is analogous to that of Theorem 3.2 in Sriram and Vidyashankar (2000). Bearing in mind Theorem 3, to obtain the eventual existence and the consistency it is enough to prove ||p 1/2 n − p 1/2 || 2 → 0, a.s. and this is shown from the convergence ofp n to p in l 1 and the inequality ||p 1/2 n − p 1/2 || 2 2 ≤ ||p n − p|| 1 . The measurability ofθ ρ n (p n ) andθ HD n (p n ) is obtained by Corollary 2.1 in Brown and Purves (1973).

Proof of Theorem 5 (i)
To prove (i) we adapt and extend the proofs of Theorem 1 in Basu, Sarkar, and Vidyashankar (1997) and of Theorem 3.4 in Park and Basu (2004) developed for general continuous models. In order to facilitate the proof, we will assume that P [Z n → ∞] = 1.
In order to establish (8.3), since it is sufficient to prove that Note that due to ∞ k=0 p ′ k (θ 0 ) = 0 and to (A4) (a), then s., with W the limit variable introduced in (A3) (this property is deduced by applying Proposition 3.5 in González, Minuesa, and del Puerto (2016)), using a central limit theorem (see Dion (1974)), one has Consequently (8.4) holds.
a bounded sequence, one obtains that there exists some K 0 ∈ R such that Hence, sup n E[A 4 n,k ] < ∞ and {A n,k } n∈N is uniformly integrable for each k ≥ 1. Therefore, since A 2 n,k = o P (1) as n → ∞, we have A 2 n,k converges to 0 in l 1 , as n → ∞.
(ii-b) The continuity of the function L → T (p(θ, α, L)) is immediate and the boundedness of the sequence {θ L } L≥0 is deduced from its convergence.
Using L'Hôpital's rule and the fact that ∞ k=0 p ′ k (θ) = 0, (8.17) is obtained. To show (8.18), we use the convergence dominated theorem in the above expression.
(ii) The proof follows similar steps to Theorem 7 in Beran (1977) and it is omitted.